Relative singularity categories

Abstract

In this thesis, we study a new class of triangulated categories associated with singularities of algebraic varieties. For Gorenstein rings, triangulated singularity categories were introduced by Buchweitz. In 2006 Orlov studied a graded version of these categories relating them with derived categories of coherent sheaves on projective varieties. This construction has already found various applications, for example in the Homological Mirror Symmetry. <br /> The first result of this thesis is a description of singularity categories for the class of Artinian Gorenstein algebras called gentle. The main part of this thesis is devoted to the study of the following generalization of singularity categories. Let X be a quasi-projective Gorenstein scheme with isolated singularities and A a non-commutative resolution of singularities of X in the sense of Van den Bergh. We introduce the relative singularity category as the Verdier quotient of the bounded derived category of coherent sheaves on A modulo the category of perfect complexes on X. We view it as a measure for the difference between X and A. The main results of this thesis are the following. <br /> (i) We prove an analogue of Orlov's localization result in our setup. If X has isolated singularities, then this reduces the study of the relative singularity categories to the affine case. <br /> (ii) We prove Hom-finiteness and idempotent completeness of the relative singularity categories in the complete local situation and determine its Grothendieck group. <br /> (iii) We give a complete and explicit description of the relative singularity categories when X has only nodal singularities and the resolution is given by a sheaf of Auslander algebras. <br /> (iv) We study relations between relative singularity categories and classical singularity categories. For a simple hypersurface singularity and its Auslander resolution, we show that these categories determine each other. <br /> (v) The developed technique leads to the following `purely commutative' application: a description of Iyama & Wemyss triangulated category for rational surface singularities in terms of the singularity category of the rational double point resolution.